Remark 3.12 The lower bounds of \(\vartheta \)  are not surprising. In fact, equation (REF ) with \(\mathfrak {L}^{\sigma ,\mu }=\Delta \)  or \(\mathfrak {L}^{\sigma ,\mu }=-(-\Delta )^\frac{\alpha }{2}\) , \(\varphi (u)=|u|^{m-1}u\) , and \(g\equiv 0\)  will have solutions which become extinct in finite time (hence, there is no conservation of mass) when \(m\in (0,\frac{(N-2)^+}{N})\)  or \(m\in (0,\frac{(N-\alpha )^+}{N})\)  respectively [1]}, [2]}. Those papers also prove that \(\vartheta \)  can reach the critical exponents \(m=\frac{(N-2)^+}{N}\)  or \(m=\frac{(N-\alpha )^+}{N}\) , but this is beyond the scope of the present paper. In effect, we include fast diffusion up to the critical exponents for both the Laplace operator and fractional Laplace like operators.

